Basic Principles of Homing Guidance
Neil F. Palumbo, Ross A. Blauwkamp, and Justin M. Lloyd
his article provides a conceptual foundation with respect to homing guid- ance upon which the next several articles are anchored. To this end, a basic geometric and notational framework is first established. Then, the well- known and often-used proportional navigation guidance concept is developed. The mechanization of proportional navigation in guided missiles depends on several factors, including the types of inertial and target sensors available on board the missile. Within this context, the line-of-sight reconstruction process (the collection and orchestration of the inertial and target sensor measurements necessary to support homing guidance) is discussed. Also, guided missiles typically have no direct control over longitudinal accel- eration, and they maneuver in the direction specified by the guidance law by producing acceleration normal to the missile body. Therefore, we discuss a guidance command preservation technique that addresses this lack of control. The key challenges associ- ated with designing effective homing guidance systems are discussed, followed by a cursory discussion of midcourse guidance for completeness’ sake.
INTRODUCTION The key objective of this article is to provide a broad The next section, Engagement Kinematics, establishes conceptual foundation with respect to homing guidance a geometric foundation for analysis that is used in the but with sufficient depth to adequately support the arti- subsequent sections of this and the other guidance- cles that follow. During guidance analysis, it is typical to related articles in this issue. In particular, a line-of- assume that the missile is on a near-collision course with sight (LOS) coordinate frame is introduced upon which the target. The implications of this and other assump- the kinematic equations of motion are developed; this tions are discussed in the first section,Handover Analysis. coordinate frame supports the subsequent derivation of
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 25 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD _ _ _ _ proportional navigation (PN) guidance. Over the past 1 = r/r is the unit vector along the LOS; v and _rLOS T 50 years, PN has proven both reliable and robust, thereby vM are the velocity_ _ vectors_ of the target and missile, contributing to its continued use. The Development of respectively;_ vR = vT – vM is the relative velocity vector; PN Guidance Law section presents the derivation of and and e is the displacement error between predicted and discusses this popular and much-used technique. The true target position. subsequent section, Mechanization of PN, discusses how As discussed, ideally, the relative velocity vector is PN may be implemented in a homing missile. Key driv- along the LOS to the true future target position (at the ers here are the type of sensor that is used to detect and time of intercept). However, Fig. 1 depicts a more real- track the target, how the sensor is mounted to the mis- istic condition where the relative velocity is along the sile, and how the LOS (rate) measurement is developed. LOS to the estimated future target position in space. For The next section, Radome/Irdome Design Requirements, this case, the interceptor will miss the target unless it briefly discusses the functional requirements imposed applies corrective maneuvers. _ on modern missile radomes. This is followed by an Figure 1 illustrates that the displacement error, e, can_ expanded section on Guidance System Design Challenges. be decomposed into two _ components: one along (e) Here, the contributors to the degradation of guidance and one perpendicular (e⊥) to the predicted target system performance (radome error and others) are dis- LOS. This decomposition is expressed in Eq. 1: cussed. Then, Guidance Command Preservation presents ee= (): 11rr a technique for commanding acceleration commands LOSLOS (1) perpendicular to the missile body that will effectively ee= = 11rr##(). maneuver the missile in the desired guidance direction. LOSLOS _ _ Although this article is mainly concerned with termi- • In this equation, x y represents_ the_ dot _(scalar)_ product nal homing concepts, the Midcourse Guidance section between the two vectors x and y, and x 3 y represents briefly discusses issues and requirements associated with the cross (vector) product between the two_ vectors. Note the midcourse phase of flight. that, because the relative velocity vector, v, is along_ the LOS to the predicted target location, the error e will Handover Analysis alter the time of intercept but does not contribute to the final miss distance. Consequently, the miss distance that When terminal guidance concepts are being devel- must be removed by the interceptor after transition to oped or related systems analysis is being performed, it _ terminal homing is contained in e⊥ (i.e., target uncer- typically is assumed that the missile interceptor is on a tainty normal to the LOS). collision course (or nearly so) with the target. In reality, Homing missile guidance strategies (guidance laws) this is not usually the case. For example, there can be dictate the manner in which the missile will guide to significant uncertainty in target localization, particu- intercept, or rendezvous with, the target. The feedback larly early in the engagement process, that precludes sat- nature of homing guidance allows the guided mis- isfaction of a collision course condition prior to terminal sile (or, more generally, “the pursuer”) to tolerate some homing. Moreover, the unpredictable nature of future target maneuvers (e.g., target booster staging events, energy management steering, coning of a ballistic mis- Actual target e sile reentry vehicle, or the weaving or spiraling maneu- position Ќ e Predicted vers of an anti-ship cruise missile) can complicate the target development of targeting, launch, and/or (midcourse) position e 1 guidance solutions that guarantee collision course con- ʈ y ditions before initiation of terminal homing. In addition, Line of sight cumulative errors are associated with missile navigation r 1x and the effects of unmodeled missile dynamics that all vR –1z add together to complicate satisfaction of a collision vM Target vT course condition. For simplicity, all of these errors can be velocity Missile velocity Missile 1 collectively regarded as uncertainties in the location of Fixed target y the target with respect to the missile interceptor. Thus, frame if the interceptor is launched (and subsequently guided 1x during midcourse) on the basis of estimated (predicted) –1z future target position, then, at the time of acquisition by the onboard seeker, the actual target position will be Figure 1. To assist in simplifying the analysis of handover to ter- displaced from its predicted position._ Figure 1 notion- minal homing, all of the contributing navigation and engagement ally illustrates this condition, where r is the LOS vector modeling errors are collectively regarded as uncertainties to the between the missile and the predicted target location; location of the target with respect to the missile at handover.
26 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE level of (sensor) measurement uncertainties, errors in ␦r 1 = ␦t the assumptions used to model the engagement (e.g., n ␦r unanticipated target maneuver), and errors in modeling ␦t missile capability (e.g., deviation of actual missile speed LOS coordinate r 1r = of response to guidance commands from the design frame r assumptions). Nevertheless, the selection of a guidance Missile r LOS strategy and its subsequent mechanization are cru- 1 = 1r ϫ 1n cial design factors that can have substantial impact on rM Target guided missile performance. Key drivers to guidance law rT design include the type of targeting sensor to be used Fixed coordinate frame (passive IR, active or semi-active RF, etc.), accuracy of 1y the targeting and inertial measurement unit (IMU) sen- sors, missile maneuverability, and, finally yet important, 1x the types of targets to be engaged and their associated maneuverability levels. We will begin by developing a –1z basic model of the engagement kinematics. This will Figure 2. In the LOS coordinate frame, the LOS rate is perpen- lay the foundation upon which PN, one of the oldest dicular to the LOS direction and rotation of the LOS takes place – and most common homing missile guidance strategies, about the 1 . is introduced. _ Engagement Kinematics: The Line-of-Sight We define this change in direction by the vector n as given in Eq. 4: Coordinate System The development presented here follows the one n / 1 . (4) given in Ref. 1. In the sequel, the following notation is t r used: X = n 3 m (read n-by-m) matrix of scalar elements _ _ Consequently, a second unit vector, 1 , is defined to x , i = 1 … n, j = 1 … m; x = n 3 1 vector of scalar ele- _ n i,j _ be in the direction of n as shown: ments x , i = 1 … n; x = n x2 = Euclidean vector i _ _ _ _ /i = 1 i _ norm of x; 1 = x/x = n 3 1 unit vector (e.g., 1 = 1) x _ x 1r/ t n in the direction of x; /t = time derivative with respect 1 ==. (5) n 1 / t n to a fixed (inertial) coordinate system; and d/dt = time r derivative with respect to a rotating coordinate system. Finally, to complete the definition of the (right-_ Consider the engagement geometry shown in Fig. 2, _ _ handed) LOS coordinate system, a third unit vector, 1, where rM and rT are the position vectors of the missile is defined as the cross product of the first two: interceptor and target with respect to a fixed coordinate ______frame of reference (represented by the triad {1x, 1y, 1z}). 1 = 1 3 1 . (6) Consequently, we define the relative position vector of r n the target with respect to the missile as shown in Eq. 2: In general, the angular velocity of the LOS coordi- _ _ _ nate system with respect to an inertial reference frame is r = r – r . (2) o oo o T M given by =+r11r n n +, 1 where the compo- _ _ nents of the angular velocity are given in Eq. 7: The relative_ position vector can be written as _r = R1r, where R = r is the target–missile range, and 1 is the o o _ _ r r = : 1r unit vector directed along r (we refer to 1r as the LOS o o n = : 1n (7) _unit vector)._ Differentiating the relative position vector, o o : r = R1r, with respect to the fixed coordinate system,_ we = 1 . obtain the following expression for relative velocity v: Thus, upon_ reexamination of Eq. 4, we note that the LOS rate, n, can be expressed as shown in Eq. 8: vr/ =+RRo 11. (3) t rrt n =+d 11o # . (8) From Eq. 3, one can see that the rate of change of the dt rr relative position vector (i.e., relative_ velocity) comprises _ two components:. (i) a change in r as a result of a change On the right-hand side of Eq. 8, the expression d1r/dt in length (R) and (ii) a change in direction (a rotation) represents the time derivative of the LOS unit vector as a result of the rate of change of the LOS unit vector. with respect to a rotating coordinate frame, and o is
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 27 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD the angular velocity of the rotating frame with respect (–aa)–: 1 = RRp o 2 to the inertial frame. The former component is equal to TM r o op zero (i.e., the LOS unit vector is a constant). Therefore, (–aaTM) : 1n =+2RR (16) oo the LOS rate and corresponding unit vectors simplify to (–aaTM).: 1 = R r the following expressions: o n1= # r Development of PN Guidance Law o # 1 (9) Many guided missiles employ some version of PN as 1 = r . n o the guidance law during the terminal homing phase. # 1r Surface-to-air, air-to-air, and air-to-surface missile Thus, from Eq. 3, the relative velocity expression is given engagements, as well as space applications (including as rendezvous), use PN in one form or another as a guid- 1–7 v1=+RRo ()o # 1 . (10) ance law. A major advantage of PN, contributing to rr its longevity as a favored guidance scheme over the last The typical guided missile control variable is intercep- five decades, is its relative simplicity of implementation. tor acceleration. Thus, taking the derivative of Eq. 10, The most basic PN implementations require low levels of and using Eq. 9, we obtain an expression for the relative information regarding target motion as compared with acceleration: other more elaborate schemes (some are discussed in the next article in this issue, “Modern Homing Missile po o opo Guidance Theory and Techniques”), thus simplifying v= R11r+ R rr+ R(#11)+ R( # rr)+ R # 1 tt ` t j onboard sensor requirements. Moreover, it has proven poopoo = R11r+2R(# rr)+R()# 11+R[#( # r)1].(1 ) to be relatively reliable and robust. As also will be seen in the next article in this issue, under certain conditions o Next,_ using the definition of in Eq. 7 and the fact and (simplifying) assumptions about target and mis- T o that 1r = [1 0 0] , we expand the term # 1r / n in sile characteristics, the PN law is an optimal guidance Eq. 11, which results in the following relation: strategy in the sense of minimizing the terminal miss distance. 11r n 1 In order to develop the PN guidance strategy, we first o oooo o # 1r ==det r n 11n – n . (12) look to the components of Eq. 16 for sufficient condi- 10 0 tions to achieve an intercept. Looking at the first com- ponent, sufficient .conditions for an intercept are i() the Here, det represents taking the determinant of a _ LOS rate is zero ( = 0), (ii) interceptor capability to matrix. From Eq. 4, the direction of n can have no com- _ . accelerate along the LOS is greater than_ or equal_ to ponent along 1 . Thus, we conclude = 0 and obtain _ _ n target acceleration along the LOS (aM • 1r aT • 1r), the result in Eq. 13: and. (iii) the initial range rate along the LOS is negative (R(0) < 0). In this case, missile-to-target_ range (R) will o # 11= o . (13) _ _ rn decrease_ _ linearly_ ((aT – aM) • 1r = 0) or quadratically (( a – a ) • 1 < 0) with respect to time and, eventually, o T M r In Eq. 13, / n . Using Eq. 13, the other cross- pass through zero. product terms in Eq. 11 yield the following results: From the discussion above, the interceptor. must accel- erate such as to null the LOS rate ( ). We look to the oo oo2 o ##()11rr=+–r 1 second component of Eq. 16 to determine how this can (14) . p p be done. We first define closing velocity asV –R. Note # 11rn= . . c ≡ that for an approaching trajectory R < 0; thus Vc > 0. Using Eqs. 13 and 14 in Eq. 11, we obtain the desired If, for the moment, we treat closing velocity and range expression for relative acceleration: as constant, then taking the Laplace transform of the second component of Eq. 16 yields the following poly- 2 ra= – a nomial in s: t2 TM
=+(–RRp o 2 )(112Ro op++RR)(oo ).1 : o rnr ((aaTMss)– ())(1n = sR –)2Vsc (). (17) (15) In Eq. 17, s represents the Laplace transform variable. From Eq. 15, the components of relative acceleration in Thus, if we define_ interceptor_ acceleration. perpendicular the LOS coordinate frame can be written as shown in to the LOS to be aM(s) • 1n = (s), then, from Eq. 17, Eq. 16: we can write the transfer function from target accelera-
28 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE tion (perpendicular to the LOS) to the corresponding As discussed, the conventional implementation of LOS rate: PN requires closing velocity and LOS rate information to produce the guidance (acceleration) commands. If we o ()s assume (for clarity) that the engagement is planar, then = 1 . (18) . ((aT s)): 1n ()sR – 2Vc + we can rewrite Eq. 20 as shown below, where l is the LOS rate in an inertial frame of reference: Referring to Eq. 18, to ensure a stable system, we require > 2V . This leads to what is known as the true . c a = NVcl . (22) PN guidance law shown in Eq. 19: Mc
o Thus, implementation of the PN guidance law in three a1M : n = NVc,.N > 2 (19) dimensions dictates the necessity to measure LOS rate in two sensor instrument axes that are mutually perpen- As is clear from the previous development, true PN com- dicular to the sensor boresight (near-coincident with the mands missile acceleration normal to the LOS. This is, measured LOS to the target). perhaps, more obvious by referring back to Eq. 13 and As mentioned previously, .the way in which closing rewriting Eq. 19 as shown in Eq. 20: velocity (Vc) and LOS rate (l) information is obtained to mechanize PN guidance (Eq. 22) is a function of the o aM = NVc # 1r,.N > 2 (20) type of target sensor that is used and how it is mounted to c the missile body. Acquiring closing velocity information _ Here, a a represents commanded missile acceleration depends primarily on the target sensor type. Given an Mc normal to the LOS. Achieved missile acceleration is (onboard) active or semi-active RF system, for example, physically realized through aerodynamic control surface the observed Doppler frequency of the target return can deflections, control thruster operation, or a combination be used to develop a good estimate of closing velocity, of both. Thus, Eq. 20 also emphasizes the fact that the Vc. In other implementations, missile-to-target range or development of PN assumes a no-lag missile response closing velocity can be periodically up-linked to the mis- (i.e., the missile is assumed to respond instantly to, and sile to facilitate PN guidance. . achieve perfectly, the guidance command). The way in which LOS rate ( l) information is derived We will simply mention another variant of PN, depends on the type of target sensor that is used and referred to a pure proportional navigation (PPN). The how it is mounted to the missile. For example, a space- names given to these variants are somewhat arbi- stabilized sensor (could be RF, IR, or laser) is mounted trary, but these names have stuck. The general three- on a gimbaled platform to increase the field of regard of dimensional version of PPN can be expressed as shown the sensor and to isolate it from missile body motion. in Eq. 21. Conversely, tracking systems that do not require a large field of regard or that employ an IR focal plane array, o for example, are fixed to the body (strapdown systems). avM = k # M . (21) c Here, we will consider space-stabilized systems. Before introducing the details on how to derive LOS rate, we Here, k is the navigation gain. Clearly, PPN commands briefly discuss space-stabilized target sensor systems. missile acceleration normal to the missile velocity _ Various space-stabilized designs are possible, but a vector, v . M typical design is one in which two mutually perpen- In view of the importance of the PN law in missile dicular gimbals are employed along with rate gyros used guidance and space applications, considerable analytical for platform stabilization and LOS/LOS rate recon- study has been conducted regarding the behavior of a struction. (Typically, these systems rely on the missile missile guided under PN. Since the differential equations autopilot for roll stabilization.) Such gimbaled platforms governing PN motion, even when considering kinemat- use a servomotor in each axis to accommodate seeker ics only, are highly nonlinear, only limited success has pointing. Hence, we will define a space-stabilized seeker been achieved in solving these equations analytically. to be composed of the target sensor (antenna/energy- collecting device and a receiver), gimbals (and associ- Mechanization of PN ated servomotors), gyros, and the necessary control elec- Here, we discuss how PN can be mechanized in a tronics. The necessary seeker functions are as follows: homing missile. Central to this topic is the type of sensor (i) track the target continuously after acquisition, (ii) that is used to detect and track the target: whether it provide a .measure of the LOS angle () or LOS angu- is a passive (e.g., IR), semi-active, or active (e.g., RF or lar rate (l), (iii) stabilize the seeker against significant laser) sensor and, as important, how it is mounted to the missile body rate motion (pitching and yawing rate) that missile. may be much larger than the LOS rate to be measured,
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 29 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD measures the tracking error ( ) with respect to seeker Measured line of sight m coordinates. (Primary contributors to tracking error are (True) line of sight bse discussed later.) The measured tracking error, in turn, Seeker centerline is used by the tracking system (the seeker track loop) to drive the seeker dish angle (via servomotor torquing m  of the gimbals) such as to minimize the tracking error, Missile body centerline thereby keeping the target in. the field of view. Conse- quently, the seeker dish rate, , is approximately equal to the inertial LOS rate. The transfer function of LOS rate to seeker dish rate can be approximated by the following m Inertial reference first-order transfer function:
Figure 3. Illustration of the two-dimensional definitions of the o 1 o = . (24) various angles often used when analyzing the LOS reconstruction l ts s + 1 process. In this relationship, ts is the seeker track-loop time con- stant. Thus, the seeker dish rate will lag the LOS rate. and (iv) measure the closing velocity, if possible.8 Item iv The accuracy to which the seeker is stabilized places is possible to achieve with radar systems but is difficult fundamental limitations on the homing precision of the with IR systems. missile. Seeker pointing is accomplished by an outer gimbal One possible LOS rate estimation scheme is shown rotation b about the missile body y axis (pitch), fol - O in Fig. 5, which illustrates a simplified block diagram lowed by an inner gimbal rotation b about the sub - I comprising the seeker, guidance computer, flight control sequent z axis. These two rotations are used to define system, and body rate aerodynamic transfer function. In the coordinate transformation from the missile body Fig. 5, the Laplace operator is indicated by s. For simplic- frame (B) to the seeker coordinate frame (S), as shown ity, the flight control system (i.e., the combined repre- in Eq. 23: sentation of the control surface actuators, aerodynamics, coscbbos sinb –sincbbos and autopilot) is expressed as the transfer function rep- O I I OI resented by G (s). The guidance system is represented CS =–cossbbin cos b sinsbbin . (2 3) FC B OI I OI as a simplified LOS rate guidance filter followed by a PN > sinbO 0 cos bO H guidance law. The combined guidance system transfer function is shown in Eq. 25, where tf is the guidance In order to outline possible approaches to derive LOS filter time constant: rate for guidance purposes, we will refer to Fig. 3, where a NV we have defined the following angular quantities: c is c = c . (25) o t the inertial angle to the missile body centerline; is the lm f s + 1 inertial angle to the seeker cen- terline (inertial dish angle); b is Noise the gimbal angle (angle between seeker boresight and missile Seeker + centerline); is the true track - + + + m ⌺ ⌺ ⌺ Receiver Tracking ing error (epsilon) between the LOS Ϸ1 error LOS and seeker centerline; – – bse  Gimbal angle 1 is a perturbation to the true epsi- s lon caused by radome refraction 1/s 1/s . c of the RF energy or irdome dis- . +  tortion of IR energy as it passes Gimbal servo Gimbal control ⌺ through the material; m is the – measured epsilon; l is the true + . Dish rate Rate gyro inertial LOS angle; and lm is ⌺ Ϸ1 the measured, or reconstructed, + . inertial LOS angle. Body attitude rate Tracking of a target requires the continuous pointing of the sensor beam at the target. As Figure 4. Simplified planar model of a gimbaled seeker track loop (without radome effects). In illustrated in Fig. 4, the receiver this configuration, the commanded dish rate is proportional to the tracking error.
30 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
Noise
Seeker LOS rate pickoff LOS rate estimate + ^. m a + + + Receiver 1 1 Mc ⌺ ⌺ ⌺ NVc LOS Ϸ1 s f s + 1 – –  Gimbal angle Guidance PN Commanded . filter acceleration c 1/s 1/s Guidance computer . +  Gimbal servo Gimbal control ⌺ Flight control GFC(s) – system + . a Achieved Dish rate Rate gyro acceleration ⌺ Ϸ1 + . A s + 1 Body attitude rate vm Body rate aerodynamics
Figure 5. Simplified planar model of a traditional LOS rate reconstruction approach that directly supplies an LOS rate mea- surement to the guidance computer. Note that the LOS rate pickoff is assumed to be proportional to the boresight error measurement. The measured LOS rate is subsequently filtered to mitigate measurement noise and then applied to the PN homing guidance law.
Moreover, the transfer function from commanded accel. - 2. Integrate the output of the missile body rate gyro, eration (from the guidance law) to missile body rate (c) obtained from the missile IMU, and sum the inte- is approximated by the following aerodynamic transfer grated IMU gyro output together with the seeker function, where tA is the turning rate time constant and gimbal angle and the measured tracking error. We vm is missile velocity: illustrate this approach in the block diagram shown co t s + 1 in Fig. 7. This approach is expressed mathematically = A . (26) as shown in Eq. 28: avcm . In this approach, the fact that the LOS rate is embedded lm = m + b + # c dt . (28) in the tracking error (m) is exploited. As illustrated, a LOS rate estimate is derived by appropriately filtering From Fig. 3, it is clear that the two concepts are algebra- the receiver tracking error scaled by the seeker track- ically equivalent in the absence of noise and assuming loop time constant. perfect instruments (no gyro biases, drift, etc.); however, Other approaches can be used to derive LOS rate for in practice, this is not the case. Moreover, we note that, homing guidance purposes; these are generally referred in general, the guidance filters for the two approaches to as either LOS reconstruction or LOS rate reconstruc- are not necessarily the same (e.g., for the simplified guid- tion. We next outline three alternative techniques (two ance filters shown in the figures, the filter time - con LOS reconstructions and one LOS rate reconstruction). stants, represented by tf , are not necessarily the same). The fundamentals of guidance filtering will be discussed LOS Reconstruction in the companion article in this issue “Guidance Filter As shown in Fig. 3, LOS reconstruction works to Fundamentals.” It is shown in Ref. 9 that the two LOS reconstruction approaches can yield significantly dif- construct a measured LOS, lm, in an inertial frame of reference. The measured LOS then is filtered (via an ferent results when noise and imperfect instruments appropriate guidance filter) to derive an estimate of LOS are used. How these differences manifest themselves rate for guidance purposes. Two different LOS recon- depends on the quality of the measurements and instru- struction approaches are as follows:9 ments that are used. 1. Integrate the seeker gyro output and sum it with the measured tracking error. A block diagram of this LOS Rate Reconstruction approach is shown in Fig. 6. Mathematically, this A guidance signal also can be generated by dif- approach can be expressed as Eq. 27: ferentiating the tracking error and adding it to . the seeker rate gyro output. For practical purposes, lm = m + # dt . (27) taking the derivative of the tracking error is accom-
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 31 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD
Noise
Seeker LOS pickoff LOS rate estimate + ^. m + a + + + Receiver m s Mc ⌺ ⌺ ⌺ ⌺ NVc LOS Ϸ1 f s + 1 – – +  Gimbal angle 1 Guidance PN Commanded s filter acceleration 1/s 1/s . Guidance computer c . +  Gimbal servo Gimbal control ⌺ Flight control GFC(s) – system + . Achieved Dish rate Rate gyro a 1/ acceleration ⌺ Ϸ1 s + . A s + 1 Body attitude rate vm Body rate aerodynamics
Figure 6. LOS reconstruction approach 1 wherein the seeker gyro output is integrated and summed with the measured tracking error to form an LOS measurement. The measured LOS is subsequently passed to the guidance computer, where it is filtered to extract an LOS rate and then applied to the PN guidance law.
Noise
Seeker LOS pickoff LOS rate estimate + ^. m + a + + + Receiver m s Mc ⌺ ⌺ ⌺ ⌺ NVc LOS Ϸ1 f s + 1 – – + 1 Guidance Commanded  Gimbal angle PN s filter acceleration . 1/s 1/s Guidance computer c . + Body attitude  Gimbal servo Gimbal control ⌺ Flight control GFC(s) – system + . Achieved Dish rate Rate gyro a 1/ acceleration ⌺ Ϸ1 s + . A s + 1 Body attitude rate vm Body rate aerodynamics
Figure 7. LOS reconstruction approach 2 wherein the output of the missile body rate gyro (via the missile IMU) is integrated and summed together with the seeker gimbal angle and the measured tracking error to develop an LOS measurement. The measured LOS is subse- quently passed to the guidance computer, where it is filtered to extract an LOS rate and then applied to the PN guidance law. plished by using a lead network such as that given as a result of the differentiation process. However, in below: Ref. 10, it is shown that LOS rate reconstruction can work well if the missile turning rate (i.e., responsiveness s . (29) to commands) is very fast. tDs + 1 Radome/Irdome Design Requirements Figure 8 illustrates this approach. Here, tD is the time constant of the lead network. As shown, the recon- In endoatmospheric engagements, a radome (or structed LOS rate is filtered by an appropriate guidance irdome) is required in order to protect the onboard filter (here represented by a simplified lag filter with time seeker from the elements. For exoatmospheric vehicles, constant tf) to derive a LOS rate estimate. This method a radome/irdome is not necessarily required. The key may lead to excessive amplification of the receiver noise dome requirements are summarized below8:
32 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
Noise
Seeker LOS rate pickoff LOS rate estimate + . . ^. m m + a + + + Receiver s m 1 Mc ⌺ ⌺ ⌺ s + 1 ⌺ NVc LOS Ϸ1 D f s + 1 – – +  Gimbal angle 1 Guidance PN Commanded s filter acceleration 1/s 1/s . . Guidance computer c . +  Gimbal servo Gimbal control ⌺ Flight control GFC(s) – system + . Achieved Dish rate Rate gyro a acceleration ⌺ Ϸ1 + . A s + 1 Body attitude rate vm Body rate aerodynamics
Figure 8. Simplified planar model of an alternate LOS rate reconstruction approach that can directly supply an LOS rate measurement to the guidance computer. Note that the LOS rate reconstruction differences between this figure and Fig. 5. As before, the measured LOS rate is subsequently filtered to mitigate measurement noise and then applied to the PN homing guidance law.
1. It must convey the energy with minimum loss. As an example, Fig. 9 illustrates three conceivable 2. It must convey the energy with minimum distortion, radome shapes. The tangent-ogive shape (on the right) particularly angular distortion because this creates a is a typical compromise design. parasitic feedback loop that can have a significant negative impact on guidance performance (dis- Guidance System Design Challenges cussed in more detail below). There are a significant number of challenges to 3. It must have minimum aerodynamic drag. designing guidance systems: the design must provide the 4. It must have satisfactory physical properties, such desired performance while remaining robust to a multi- as sufficient strength, resistance to thermal shock tude of error sources, limited control system bandwidth, (from rapid aerodynamic heating), resistance to and inherent system nonlinearities. Some of these chal- rain erosion at high speeds, and minimum water lenges are summarized below. absorption. 1. The root-mean-square final miss distance from all deleterious noise sources must be minimized. 1/2 1 2. Guidance system stability must be maintained in the parasitic feedback loop (as mentioned above, this is caused by angular distortion of the radome/irdome). L/D Х 1/2 3 Ideal electromagnetically Radome angular distortion, in particular, is a key 1 contributor to final miss distance but is considered 5 separately from other noise sources as its impact on L/D Х 3 1 Compromise radome guidance system stability is substantial. 3. System nonlinearities must be avoided as much L/D Х 5 as possible; e.g., seeker gimbal angle and gyro rate Ideal aerodynamically saturations must be avoided, as should command- ing missile acceleration beyond what is physically Figure 9. Three possible radome shapes are illustrated. For realizable by the missile. minimum angular distortion, a hemispherical shape (or hyper- hemispherical shape as in a ground-based radar) would be ideal electromagnetically (upper left), but the drag penalty is excessive. Contributors to Final Miss Distance From an aerodynamic perspective, the lower left radome shape is With respect to guidance challenge item 1 above, preferable, but it tends to have significant angular distortion char- there are a number of contributors to final miss distance acteristics. The tangent-ogive shape (on the right) is a typical com- (other than angular distortion of the radome/irdome). promise design. Nevertheless, some missiles use much blunter For example, in either an RF or IR guidance system, mea- dome designs despite the drag penalty. L/D, lift-to-drag ratio. surement noise from the various instruments (onboard
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 33 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD seeker, IMU, etc.) will tend to increase the final miss dis- 6. Initial heading errors at the start of homing. Head- tance. In a radar-guided system, high illumination power ing error can be defined as the angle between the on the target is desired in order to reduce the necessary actual missile velocity vector at the start of terminal receiver gain and, consequently, the internally gener- homing and the velocity vector that would be nec- ated noise. Similarly, a passive IR seeker is designed to essary to put the missile on an intercept course with have a maximum aperture size to maximize incoming the target. Heading error can be viewed as an initial power. Therefore, antenna/aperture size usually is made condition disturbance to the guidance system at the a maximum within the constraint of missile body diam- start of terminal homing. eter to maximize power reception and minimize angular 7. Target acceleration (maneuver) perpendicular to the beam width, thereby leading to less noise. LOS. A target can maneuver for any number of rea- The ability for any given target sensor to resolve sons (to avoid detection, to enact necessary course closely spaced objects also is limited, contributing to corrections, to evade a pursuing interceptor missile, additional miss distance. For example, a missile radar etc.). Whatever the reason, from the viewpoint of antenna usually has a relatively wide beamwidth, and so the guided missile, target maneuver can have a sto- it is unable to resolve closely spaced targets in angle until chastic quality to it and, in some instances, it can very late in the endgame of the engagement. In this sort induce very large final miss distance. Relevant target of ambiguous situation, the homing missile can incur a maneuver (acceleration) levels and corresponding large final miss distance. characteristics (horizontal weave, corkscrew, hard- Some of the key contributors to final miss distance turn, etc.) are key drivers to the design of missile 8, 11 are introduced below : guidance systems. Target maneuver generally is con- sidered a (potentially significant) disturbance rather 1. Seeker receiver range-dependent angle noise, rd. This noise is a function of the strength of the target than noise to the guidance system. return power and, hence, the signal-to-noise ratio. Items 1–5 are stochastic noise sources, and the Range-dependent noise power varies inversely with remaining items typically are referred to as non-noise 2 range to target (R) as 1/R for semi-active radar contributors to final miss distance. and as 1/R4 for active radar. For IR systems, range- dependent angle noise typically is not considered to Radome/Irdome Refraction Error be a primary contributor. The ogival shape of radomes helps to reduce drag 2. Seeker range-independent angle noise, sri. This but is a significant source of trouble to guidance system noise is independent of target return power and is performance. Irdome shapes tend to be less problematic caused by a number of internal sources like signal than radome shapes, but their effect on guidance system processing and quantization effects, gimbal servo performance still must be considered. The fundamental drive errors, and other electrical noise. issue is that as radiation passes through the dome, it is refracted by a certain angle which, in turn, depends on 3. Scintillation (and glint) noise, sglint. These are cou- pled effects and are caused by target reflections that the seeker look angle, b. As illustrated in Fig. 10,11 the vary in amplitude and phase over time. (The effect radome distorts the boresight error measurement by an is much like that of sunlight glinting from the shiny amount bse. The magnitude of refraction during homing surfaces of an automobile.) Scintillation/glint noise depends on many factors, including dome shape, fine- can be very severe and is a function of the physical ness ratio, thickness, material, temperature, operating dimensions and motion of the target. frequency, and polarization of the target return signal. Hence, it is very difficult to compensate perfectly for 4. Clutter and multipath noise, s . These noise effects c dome error a priori. Typically, the specification of dome become important at low altitudes, over either the characteristics involves the radome/irdome slope r (see land or sea. This noise is caused by the unwanted Fig. 10), which is a local property of the dome and is scattering (forward and backward) of radar returns defined as shown in Eq. 30: from, for example, sea-surface waves. 5. Imperfect seeker stabilization caused by seeker gyro 2 r / bse . (30) and gimbal scale factors, gyro drift, and missile body 2 bending (flexing). Imperfect seeker stabilization can introduce parasitic feedback loops similar to that Although r is not a constant over the entire dome, it caused by radome/irdome distortion. The net effect can be regarded as constant over a small range of seeker can be viewed as inducing a bias or perturbation on look angles, b. The most optimistic situation is one in the measured LOS rate. Exactly how these errors which the designer would be able to specify the manu- affect guidance performance will be a function of the facturing tolerances and, thus, the limits on the allow- way in which LOS reconstruction is mechanized.9 able variations of r.
34 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
bse Apparent LOS Using the fact that the seeker dish angle can be written (radome error) True LOS as = c + b, we obtain the following expression for the derivative of the measured tracking error: Seeker centerline . . . . m = l – (1 – r)b – c . (33)  (Seeker gimbal angle)
Missile body axis Using Eq. 33, we can modify the block diagram shown in Fig. 5 to include the dome distortion effect (via Eq. 33). The result is shown in Fig. 11. Clearly, the boresight error Radome creates another signal path in the missile guidance loop. bse To obtain an alternate view of this parasitic loop Ѩbse r ϵ (slope) and how it can affect guidance performance, we revisit Ѩ Eq. 32. Referring to Fig. 3, we note that the seeker gimbal angle can be expressed as b = – c and that the measured LOS can be written as lm = + m. Also noting that it is typical for r << 1, we can approximate  Eq. 32 as . . . lm ≅ l – r c . (34)
Figure 10. This illustration highlights the fact that the missile Continuing, we also require transfer function descrip- radome (or irdome) distorts the boresight error measurement. The tions of all components represented in Fig. 5. An approx- boresight error distortion is a function of look angle. imate representation of the seeker block illustrated in Fig. 5 is given by the transfer function Gs(s) below:
Referring once again to Fig. 3, it follows that we can ss () s Gss ()_ = . (35) write the following angular relationship for the mea- l()s ts s + 1 sured tracking error: Directly from Fig. 5, the combined guidance system m = 1 bse – . (31) (filter and law) transfer function can be expressed as
Differentiating Eq. 31 and using the definition of radome as() NV Gs()_ Mc ==NV Gs() c . (36) slope shown in Eq. 30, we obtain the following expression g sslt() cF s + 1 . f for m: Next, we assume the transfer function from com- ...... m = l + bse – = l + rb – . (32) manded to achieved missile acceleration, GFC(s), can be
Noise
Seeker LOS rate pickoff LOS rate estimate + ^. m a + + + Receiver 1 1 Mc ⌺ ⌺ ⌺ NVc LOS Ϸ1 s f s + 1 – – Gimbal +  bse Guidance PN Commanded angle Boresight error . r filter acceleration 1/ c s 1/s Radome Guidance computer slope . +  Gimbal servo Gimbal control ⌺ Flight control GFC(s) – system + . a Achieved Dish rate Rate gyro acceleration ⌺ Ϸ1 + . A s + 1 Body attitude rate vm Body rate aerodynamics
Figure 11. Simplified planar model of a gimbaled seeker track loop with the inclusion of radome/irdome distortion effects on the measured boresight error.
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 35 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD approximated by the following first-order lag representa- engagements), then the effective navigation ratio Ñ → 2. tion: Recall from the discussion of PN that a navigation gain of 2 or less will lead to guidance instability. This asM() 1 GsFC ()_ = . (37) fact is illustrated in Fig. 13 for N = 3 and three differ- asMc () tFC s + 1 ent ratios of Vc/vM. Conversely, when r is negative, the Finally, recall that the missile aerodynamic transfer func- effective navigation ratio Ñ will increase, resulting in additional noise throughput, which also can have detri- tion from acceleration to body rate can be approximated mental effects on guidance performance. as shown in Eq. 26 and repeated here for convenience: Dynamic Effects ssc() tAs + 1 GsA()_ = . (38) It is shown in Ref. 4 that the guidance system transfer asMm() v function given by Eq. 39 and illustrated in Fig. 12 can be Using Eqs. 34–38, Fig. 5 can be drawn as illustrated in approximated further by the following simplified transfer Fig. 12. Referring to this block diagram, the closed-loop function: transfer function from LOS rate to missile acceleration can be written as asMc() NV 1 GsR ()_, . (41) ssl() 11++r stR as() Gs()_ M R l ss() In Eq. 41, r = rNVc/vM, and the approximate overall NV Gs()Gs()Gs()/s (39) time constant, t , is given by the composite expression = cF FC s . R rNVG ()sG()sG ()sG()s in Eq. 42: 1 + cA FFCs svm ()ttsf++trFC + tA t = . (42) Static Effects R 1 + r Without refraction (r = 0), the steady-state gain of the system in Eq. 39 is given by Referring to Eqs. 41 and 42, when the dynamic pres- sure is low (e.g., high altitudes and/or low missile veloc- gG_ lim ()sN= V , ity), the aerodynamic time constant t tends to take on ss s" 0 Rcr = 0 A relatively large values (the missile turning rate becomes where recall that N > 2. On the other hand, if r ≠ 0, sluggish). For positive values of r, this effect is exacer- then the steady-state gain of the system is given as bated and the missile guidance loop becomes even more sluggish. On the other hand, when the dynamic pressure NV is low, large negative values of r coupled with large t gG_/lim ()s = c Nru ()V . (40) A ss s" 0 R r ! 0 NV c can induce guidance instability. 1 + r c vM As an example, consider Fig. 14, which illustrates the deleterious effects of radome-induced boresight error on In Eq. 40, we have defined the resulting effective homing guidance performance. This example uses PN navigation ratio as Ñ(r) = N/(1 1 rNVc/vM). Suppose guidance to guide the missile against a non-maneuvering the navigation ratio of the system is N = 3 (for the r = 0 target. No other noise sources are considered other than case). Then, examining Ñ(r), if r is positive and (rela- radome boresight error. Several curves are shown, each tively) large and the ratio Vc/vM increases beyond unity representing an autopilot response capability given by (as is often the case, particularly for near head-on the time constant t = tFC. Clearly, a more sluggish time constant exacerbates the effects of radome error slope on homing Seeker Guidance Guidance Flight dynamics filter law control performance. + s s 1 s ˆ a 1 a m NV Mc M LOS ⌺ s + 1 s + 1 c s + 1 Guidance System Nonlinearities _ s f FC In a guided missile, there are limits to the region of linear opera- s A s + 1 tion. For example, the lateral accel- r 1/s v m eration that a guided missile can Radome slope Body rate attain is limited in one of two ways: aerodynamics (i) for low-altitude intercepts, struc- Figure 12. Alternate simplified block diagram of a (linear) missile guidance and control tural considerations will limit the system that accounts for seeker dynamics, radome distortion, guidance filter effects, and maximum acceleration levels, and autopilot/airframe response characteristics. (ii) for high-altitude intercepts, the
36 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
Navigation ratio: N = 3 that the flight control system Effective navigation ratio: Ñ (r) = N/(1 + rNVc /vM) (autopilot) must be designed to 0.35 produce minimum overshoot to an acceleration step command. 0.30 If/when the guidance command
Unstable region: Ñ (r ) ϵ N/(1 + rNVc /vM) Յ 2 saturates, the missile guidance 0.25 loop is essentially opened. If guid-
M ance command saturation persists /v c 0.20 long enough, and if this persistent
rV Boundary = 0.17 saturation occurs near intercept, a 0.15 = 3 significant final miss distance can /v M Vc = 2 result. /vM 0.10 Vc Missile seeker systems also are = 1 Vc /vM prone to nonlinear saturation, 0.05 which can have significant del- Stable region: Ñ (r ) ϵ N/(1 + rNVc /vM) > 2 eterious effects on overall guid- 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 ance performance. For example, Radome slope r (rads) on gimbaled platforms, the seeker gimbal angle can saturate under Figure 13. For a navigation ratio of 3, the effect of positive radome boresight error slope r( ) certain stressing conditions (e.g., for increasing ratios of closing velocity over missile velocity is shown. To maintain guidance while pursuing a highly maneu- stability, the effective navigation ratio must remain greater than 2. vering target). If this saturation occurs, the guidance loop has = 0.5 s–1 = 0.3 s–1 = 0.1 s–1 effectively become open and, if this occurs near inter- = 0.4 s–1 = 0.2 s–1 cept, significant final miss distance can occur. 10 9 Guidance Command Preservation 8 7 Three-dimensional guidance laws usually generate a guidance command vector without consideration to 6 how the missile interceptor will effectuate the specified 5 maneuver. For example, it is typical for a guided missile 4 to (i) have no direct control over its longitudinal accel- 3 eration (e.g., axial thrust is not typically throttleable) 2 and (ii) maneuver in the guidance direction (specified Closest point of approach (ft) by the guidance law) by producing acceleration normal 1 to the missile body. Therefore, the question naturally 0 arises as to how a three-dimensional guidance com- –0.10 –0.05 0 0.05 0.10 0.15 Radome error slope (%) mand, specified by the guidance law without con- sideration of the aforementioned constraints, can be Figure 14. This plot illustrates the miss performance of PN achieved by maneuvering the missile perpendicular to versus radome boresight error slope. The results were derived by the missile centerline. We shall refer to this process as using a planar Simulink model assuming no error sources aside guidance command preservation. from radome error. Each curve represents the results using an In Ref. 1, two guidance command preservation autopilot with specified time constant t. A larger time constant techniques are developed. The first approach, referred indicates a larger lag in the missile response to acceleration to here as GCP1, is further elaborated in Ref. 12. The commands. second guidance command preservation technique (GCP2) is a minimum norm solution. As such, GCP2 could (theoretically) provide savings in commanded acceleration limit is more a result of maximum angle-of- acceleration (or fuel use in exoatmospheric applica- attack limitations. In either case, the missile guidance tions) as compared with the first method. The GCP1 and control system design must take the acceleration technique is discussed here. limits into account. The most significant implication is that the commanded acceleration from the guid- GCP1 Technique ance system must be hard-limited so as not to exceed To describe this guidance command preservation the acceleration limits of the missile, which also implies technique, we refer to Fig. 15 and make the following
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 37 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD
1 LOS Equation 45 is derived independently from any particu- r lar coordinate system. It is easy to mechanize the expres- 1x Missile sion in Eq. 45 such that the computations are carried out centerline in the missile body frame (B). We first note that missile ʈa ʈ = cos–1 c ʈa ʈ acceleration resolved into the missile body frame can M B B B B T be expressed as aM = []aaMx My aMz and that the body-referenced unit vector along_ the longitudinal axis B T Missile of the missile_ can be written as 1x =_ [1 0 0] . The acceleration aMЌ a PN guidance command Ќ LOS Ќ body c quantities 1r (LOS unit vector) and ac (guidance law aM Achieved missile acceleration command) usually are defined with respect to a guid- ance reference_ frame. To reflect this_ fact, we express Figure 15. GCP1. The LOS vector is shown along with the com- G T G G G T them as 1r = [1 0 0] and ac = [0 acy acz] . manded acceleration from PN, which is perpendicular to the LOS. Here, the superscript denotes the frame of reference Also, the missile centerline (direction of the missile nose) and with which the quantities are currently expressed; in achieved missile acceleration vector are superimposed. The guid- this case, the guidance reference frame (G). Thus, we ance preservation problem is to achieve missile acceleration that must resolve all guidance frame quantities into the has the component along the guidance direction as specified by missile body frame through the coordinate transfor_ - the guidance law. B B B mation CG = cG(i, j), i, j = 1, 2, 3; for example, 1r = B T B B B T CG[1 0 0] ≡ [cG(1, 1) cG(2, 1) cG(3, 1)] . Note that _ if we fix the guidance frame at the start of terminal definitions: 1 is a unit vector along the missile–target G _ r homing (t = 0), we obtain the transformation CI = LOS, 1 is a unit vector along the missile center- B S B x _ CI (CBt = 0). The inertial-to-body transformation, CI , line (missile longitudinal axis), a is the guidance law S c comes from the IMU, and CB may be computed via the acceleration command perpendicular to the LOS, and seeker gimbals (see Eq. 23). Subsequently, CB = CB(CG)T. _ T G I I aM = [aMx aMy aMz] is the achieved missile accelera- If necessary, the guidance frame may be updated from tion vector comprising components in the x/y/z axes. instant to instant, but this adds computational complex- Two component directions serve as a basis for the B _ ity. Given that CG is available, the following mechaniza- development of GCP1: the direction of the_ LOS_ _ (1r) tion of Eq. 45 is possible: and the commanded guidance direction (1 = a /a ). ac c c Assuming perfect interceptor response to commanded aB – a1B : B acceleration (i.e., no lag from commanded to achieved aB =+Mx c x raB B GC B B c acceleration), the (guidance preserved) missile accelera- f r1: x p R V R V tion can be expressed as follows: cB (,11) aB (46) aaB – B S G W S cxW / Mx cx S B W S B W _ _ _ B cG (,21) + acy . c (,11) S B W S W aM = k11r + k21a . (43) f G pS W B c cG (,31) SaczW T X T X The design parameters, k1 and k2 , are determined sub - B B B T In Eq. 46, [acx acy acz] is the guidance command ject to the following constraints: vector expressed in the missile body frame. This mecha- nization will generate guidance commands in the missile a1M : x = aMx body frame that satisfy the constraints in Eq. 44. (44) a1: = a . Mcac
Equations 44 analytically embody the following con- MIDCOURSE GUIDANCE straints: (i) the missile does not control longitudinal So far, this article has been chiefly concerned with the requirements of terminal homing, wherein target acceleration and (ii) the component _ of missile accel- eration along the guidance direction, 1 , must be that measurements are provided by one or more onboard ter- _ ac minal sensors and minimizing miss distance at intercept specified by the_ _ guidance law (ac). Using Eqs. 43 and 44 and given 1 • 1 = 0 (i.e., the guidance command is the primary objective. For completeness’ sake, we now r ac is perpendicular to the LOS), an expression for com- will briefly discuss issues and requirements associated manded missile acceleration that will preserve the guid- with the midcourse phase of flight. ance command is given in Eq. 45: During the midcourse guidance phase of a multi- mode missile (see Fig. 16), target tracking is performed by an external sensor to support the engagement.6 Exter- : aMx – a1c x nal tracking relaxes the requirements for the onboard aGL =+1ar c . (45) eo11r : x sensor to point at the target and detect it at large ranges.
38 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
Boost (Inertial) Midcourse (Guided) Terminal (Guided) Predicted intercept point (PIP) • Onboard inertial guidance processing EOB • Safe launch/separation In-flight • Boost to flight speed target • Establish flight path updates • Arrive at pre-calculated point Handover Endgame Fuzing at end of boost (EOB) Seeker acquisition
• Off-board target tracking • Onboard seeker/guidance processing • Onboard or off-board guidance processing • Requires high degree of accuracy and fast • One or more additional booster stages command response • Maintain a desired course • Can require maneuvering to maximum • Bring the missile “close” to the target capability to intercept fast-moving, • Trajectory shaping/energy management evasive targets • Can be active endo through exo • Active endo or exo, typically not both Pre-launch • Targeting• Targeting • Engageability• Engageabi ldeterminationity determination • Missile• Mi sinitializationsile initialization
Figure 16. Missile guidance phases. The weapon control system first decides whether the target is engageable. If so, a launch solution is computed and the missile is initialized, launched, and boosted to flight speed. Inertial guidance typically is used during the boost phase of flight during which the missile is boosted to flight speed and roughly establishes a flight path to intercept the target. Midcourse guid- ance is an intermediate flight phase whereby the missile receives information from an external source to accommodate guidance to the target. During the midcourse phase, the missile must guide to come within some reasonable proximity of the target and must provide desirable relative geometry against a target when seeker lock-on is achieved (just prior to terminal homing). The terminal phase is the last and, generally, the most critical phase of flight. Depending on the missile capability and the mission, it can begin anywhere from tens of seconds down to a few seconds before intercept. The purpose of the terminal phase is to remove the residual errors accumulated during the previous phases and, ultimately, to reduce the final distance between the interceptor and target below some specified level.
However, the accuracy of sensors generally degrades with A form of PN might be used during the midcourse distance. It is therefore unlikely that a standoff sensor guidance phase but generally results in excessive slow- will have sufficient accuracy in tracking both the target down, however, as a result of the added atmospheric and missile to guide a missile close enough for intercept. drag generated by maneuvers. Instead, it is desirable to The objectives during midcourse guidance are instead use a guidance law that will maximize missile velocity to guide the missile to a favorable geometry with respect during the endgame such that missile maneuverabil- to the target for both acquisition by the onboard sensor ity will be maximized when called for during stressing and handover to terminal homing. endgame maneuvers. Lin11 describes an approach that During the midcourse guidance phase, the weapon applies optimal control theory to derive efficient, ana- control must provide information to the missile about lytical solutions for a guidance law that approximately the threat. This information may be nothing more than maximizes the terminal speed while minimizing the estimates of the threat kinematic states or may include a miss distance. The guidance commands are expressed in predicted intercept point (PIP). Predictions of the future the form given in Eq. 47: target trajectory, whether calculated on board the missile or by the weapon control system, are based on assump- K K a = 1 V2 sinc()os()–(2 V2 sin ). (47) tions of what maneuvers the threat is likely to do and R R what maneuvers are possible for the threat to perform. These assumptions typically are sensitive to the type of Here, V is the missile speed, R is the range to the threat assumed but may include booster profiles, aerody- PIP, is the angle difference between the current and namic maneuverability, or drag coefficient(s). The PIP desired final velocity, and s is the heading error. The is therefore highly prone to errors. Occasional in-flight first term shapes the trajectory to achieve a desired target updates may be needed to improve the accuracy of approach angle to the intercept, and the second term the PIP before handover to terminal homing. minimizes miss. The gains K1 and K2 are time-varying
JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) 39 N. F. PALUMBO, R. A. BLAUWKAMP, and J. M. LLOYD and depend on flight conditions, booster assumptions, missile applications, with a focus on LOS reconstruction and other factors. A critical component of implementing and guidance command preservation. With respect to this law is computing an accurate estimate of time-to-go homing guidance, we itemized the primary contribu- when the missile is still thrusting, which depends on the tors to guidance performance degradation that can ulti- component of missile acceleration along the LOS. The mately lead to unacceptable miss distance. guidance calculations can be made on board the missile The onboard missile seeker has a limited effective or off board, with acceleration commands passed from range beyond which target tracking is not possible. To the weapon control system to the missile where they are support engagements that initially are beyond such a converted to body coordinates. range, midcourse guidance is used to bring the missile For exoatmospheric flight, drag is no longer an issue, within the effective range of the seeker. Thus, in con- but long flight times will result in curved trajectories as trast to terminal homing, during the midcourse guid- a result of gravity. From the current position, the Lam- ance phase of flight, the target is tracked by an external bert solution (see Ref. 7 for a simple synopsis) defines sensor and information is uplinked to the missile. The the necessary current velocity to reach a terminal key objectives during midcourse guidance are to guide intercept point at a given time. A guidance law can be the missile to a favorable geometry with respect to the “wrapped around” the Lambert solution to progressively target for both acquisition by the onboard missile target- steer the missile velocity vector to align with the Lam- ing sensor and to provide acceptable handover to ter- bert velocity. Then either thrust termination or a slow- minal homing. Many of the terminal homing concepts down maneuver can be used to match the magnitude of discussed here and in the subsequent articles on modern the velocity of the Lambert solution when the missile guidance and guidance filtering in this issue also are booster burns out. applicable to developing midcourse guidance policies. Thus, mainly for completeness, we briefly introduced the problem of midcourse guidance. CLOSING REMARKS The key objective of this article was to provide a REFERENCES relatively broad conceptual foundation with respect 1Pue, A. J., Proportional Navigation and an Optimal-Aim Guidance Tech- to homing guidance but also of sufficient depth to nique, Technical Memorandum F1C(2)80-U-024, JHU/APL, Laurel, MD (7 May 1980). adequately support the articles that follow. First, we 2Ben-Asher, J. Z., and Yaesh, I., Advances in Missile Guidance Theory, discussed handover analysis and emphasized that the American Institute of Aeronautics and Astronautics, Reston, VA (1998). displacement_ error between predicted and true target position, e, can be decomposed into two components: 3Locke, A. S., Principles of Guided Missile Design, D. Van Nostrand _ _ Company, Princeton, NJ (1955). one along (e ) and one perpendicular to (e ) the pre- ⊥ 4Shneydor, N. A., Missile Guidance and Pursuit: Kinematics, Dynamics dicted LOS. Thus, because the relative velocity is and Control, Horwood Publishing, Chichester, England (1998). along the LOS to the predicted target location, the error 5Shukala, U. S., and Mahapatra, P. R., “The Proportional Navigation along this direction alters the time of intercept but does Dilemma—Pure or True?” IEEE Trans. Aerosp. Electron. Syst. 26(2), not contribute to the final miss distance. It is the error 382–392 (Mar 1990). 6Witte, R. W., and McDonald, R. L., “Standard Missile: Guidance perpendicular to the LOS that must be removed by System Development,” Johns Hopkins APL Tech. Dig. 2(4), 289–298 the interceptor after transition to terminal homing to (1981). effect an intercept. 7Zarchan, P., Tactical and Strategic Missile Guidance, 4th Ed., American Next, we developed a classical form of PN and noted Institute of Aeronautics and Astronautics, Reston, VA (1997). 8Stallard, D. V., Classical and Modern Guidance of Homing Interceptor that a primary advantage of PN, contributing to its lon- Missiles, Raytheon Report D985005, presented at an MIT Dept. of gevity as a favored guidance scheme over the last five Aeronautics and Astronautics seminar (Apr 1968). decades, is its relative simplicity of implementation. In 9Miller, P. W., Analysis of Line-of-Sight Reconstruction Approaches for fact, the most basic PN implementations require low SM-2 Block IVA RRFD, Technical Memorandum F1E(94)U-2-415, JHU/APL, Laurel, MD (22 Dec 1994). levels of information regarding target motion as com- 10Nesline, F. W., and Zarchan, P., “Line-of-Sight Reconstruction for pared with other, more elaborate schemes, thus sim- Faster Homing Guidance,” AIAA J. Guid. Control Dyn. 8(1), 3–8 plifying onboard sensor requirements. Moreover, it has (Jan–Feb 1985). proven to be relatively reliable and robust. This particu- 11Lin, C. F., Modern Navigation, Guidance, and Control Processing, Pren- tice Hall, Englewood Cliffs, NJ (1991). lar (and somewhat unique) treatment of PN was taken 12 1 Abedor, J. L., Short Range Anti-Air Warfare Engagement Simulation, from a 1980 APL memorandum written by Alan J. Pue. Technical Memorandum F1E(86)U-3-031, JHU/APL, Laurel, MD We also discussed how PN can be mechanized for guided (10 Nov 1986).
40 JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 29, NUMBER 1 (2010) BASIC PRINCIPLES OF HOMING GUIDANCE
Neil F. Palumbo is a member of APL’s Principal Professional Staff and is the Group Supervisor of the Guidance, TheNavigation, and ControlAuthors Group within the Air and Missile Defense Department (AMDD). He joined APL in 1993 after having received a Ph.D. in electrical engineering from Temple University that same year. His interests include control and estimation theory, fault-tolerant restructurable control systems, and neuro-fuzzy inference systems. Dr. Palumbo also is a lecturer for the JHU Whiting School’s Engineering for Professionals program. He is a member of the Institute of Electrical and Electronics Engineers and the American Institute of Aeronautics and Astronautics. Ross A. Blauwkamp received a B.S.E. degree from Calvin College in 1991 and an M.S.E. degree from the University of Illinois in 1996; both degrees are in electrical engineering. He is pursuing a Ph.D. from the University of Illinois. Mr. Blauwkamp joined APL in May 2000 and currently is the supervisor of the Advanced Concepts and Simulation Techniques Section in the Guidance, Navigation, and Control Group of AMDD. His interests include dynamic games, nonlinear control, and numerical methods for control. He is a member of the Institute of Electrical and Electronics Engineers and the American Institute of Aeronautics and Astronautics. Justin M. Lloyd is a member of the APL Senior Professional Staff in the Guidance, Navigation, and Control Group of AMDD. He holds a B.S. in mechanical engineer- ing from North Carolina State University and an M.S. in mechanical engineering from Virginia Polytechnic Insti- tute and State University. Currently, Mr. Lloyd is pursuing his Ph.D. in electrical engineering at The Johns Hopkins University. He joined APL in 2004 and conducts work in optimization; advanced missile guidance, navigation, and con- trol; and integrated controller design. For further information on the work reported here, contact Neil Palumbo. His email address is neil.palumbo@ Neil F. Palumbo Ross A. Blauwkamp Justin M. Lloyd jhuapl.edu.
The Johns Hopkins APL Technical Digest can be accessed electronically at www.jhuapl.edu/techdigest.
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